(a) If the order of the choices matters, this is a permutation problem. We want to choose 4 objects from 17 distinct objects without replacement. The number of ways to do this can be calculated using the formula for permutations:
P(n, r) = n! / (n - r)!
where n is the total number of objects and r is the number of objects to be chosen.
In this case, we have 17 objects and we want to choose 4, so the number of ways is:
P(17, 4) = 17! / (17 - 4)! = 17! / 13! = 17 × 16 × 15 × 14 = 43,680.
Therefore, there are 43,680 ways to choose 4 objects without replacement if the order of the choices matters.
(b) If the order of the choices does not matter, this is a combination problem. We want to choose 4 objects from 17 distinct objects without replacement. The number of ways to do this can be calculated using the formula for combinations:
C(n, r) = n! / (r! * (n - r)!)
In this case, we have 17 objects and we want to choose 4, so the number of ways is:
C(17, 4) = 17! / (4! * (17 - 4)!) = 17! / (4! * 13!) = (17 × 16 × 15 × 14) / (4 × 3 × 2 × 1) = 2380.
Therefore, there are 2380 ways to choose 4 objects without replacement if the order of the choices does not matter.